Mean-field Stochastic Maximum Principle For Optimization With Recursive Utilities

Peng[1]in 1998 raised an important open problem:"The corresponding 'global maximum principle' for the case where f depends nonlinearly on z is open,except for some special case".Then this problem was studied in[2]and[3].But the maximum principles obtained in[2]and[3]still contain some unknown parameters.The main dif-ficulty arises when the generating function f(x,y,z,u)of the BSDE depends nonlinearly on z,which is,practically and theoretically,a typical situation in such problem.This longtime standing open problem has been solved by Hu[4]in 2015.Base on Hu[4],this paper generalizes this result to the case of mean-field,and also considers the fully cou-pled and partially observed mean-field stochastic optimal control problem.This paper mainly has two parts.In the first part,we study the mean-field maximum principle for a stochastic opti-mal control system with recursive cost functional.We obtain the variational equations for mean-field backward stochastic differential equation(MFBSDE,in short)in recursive stochastic optimal control problems,and then get the maximum principle which is new.The control domain need not to be convex,and the generator of MFBSDE can contain z.We consider the following state equation:Our aim is to get the condition which the optimal control u(·)needs to satisfy,minimizing the cost functional.Our main idea is that using Ekeland's variational princi-ple and adjoint equations,we obtain the necessary condition for getting optimal control maximum principle.The main difficulties of this part are how to get the specific form of the second order equation of the MFBSDE(which is different to[15]),and the com-plexity of the second order adjoint equation due to the quadratic form of the variational equation on z.In the second part,we study partially observed stochastic optimal control prob-lems for fully coupled forward-backward stochastic systems.By Ekeland's principle and spike variation method,the Pontrygin's maximum principle is obtained in global form under the assumption that the forward diffusion coefficient does not contain the control variable and the control domain is not necessarily convex.And the related adjoint pro-cesses are characterized as solutions of related forward-backward stochastic differential equations(FBSDE,in short).However,the general stochastic principle of fully coupled forward-backward optimal control problem is still an open problem.We define the following cost functional:The equation(5)exists a unique solution under the Lipschitz condition,integrability condition and G-monotonic condition.By Ekeland's variational principle,we obtained the mean-field maximum principle.